The weighted $k$-path vertex cover problem on series-parallel graphs

Given a graph $G$ with a vertex weight function $\omega_V:~V(G)\to\mathbb{R}^+$ and a positive integer $k,$ we consider the weighted $k$-path vertex cover problem: it consists in finding a minimum-weight subset $S$ of vertices of a graph $G$ such that every path of order $k$ in $G$ contains at least one vertex from $S.$ We give $O(n)$ algorithms for finding the minimum weight of $k$-path vertex cover and connected $k$-path vertex cover for series-parallel graphs.